RATIO. 



logarithmic fyftcm is .43429448, which is the reciprocal of 

 2.3025809,1110 hypcrboHclogarithm of 10, the radix of this 

 fyltem. 



It is fliewn under the article Logarithms, that 



log. a = 



(«- I) -I ja-i y + i(a-i )', &c. 

 (r- I) -i(/-- 1)^+ J(r- l)',&c. 



where r is the radix, and may be affumed at pleafure, and 

 the reciprocal of this whole denominator is called the mo- 

 dulus. In the hyperbolic fyllem the whole denominator 

 is alTumed l, which makes r = 2.71828182 ; and in the 

 common fyllem r is affumed 10, and the whole feries becomes 

 — 2. 3c 25 809, which is tlie reciprocal of the modulus ; and 

 fince, in the former cafe, we have 



hyp. log. a = {a - 1) - i (a - ly + Ha - i)\ &c. 



it follows alfo that in any other 



hyp. log. r = (r — i) 



X(r- ir + ; (/■- 1)^&C. 



whatever be the value of r ; therefore, in every fyllem, the 

 modulus is the reciprocal of hyp. log. of the radix. See 



LOGAUITHMS. 



Ratio, Denominator of a. See Denominator. 



Ratio, Properties of.— l. Ratios fimilar to the fame 

 third are alfo fimilar to one another ; and thofe fimilar to 

 fimilar, are alfo fimilar to one another. 



2. If A : B :: C : D ; then, inverfely, B : A :: U : C. 



3. Similar parts P and p have the fame ratio to the 

 wholes T and t ; and if the wholes have the fame ratio, the 

 parts are fimilar. 



4. If A : B :: C : D ; then, alternately, A : C :: B : 

 D. And hence, if B - D, A ^ C ; hence, alfo, if A 

 : B :: C : D ; and A : F :: C : G ; we (hall have B : F :: 

 D : G. Hence, again, if A : B :: C : D ; and F : A :: 

 G : C ; we (hall have F : B :: G : D. 



5. Thofe things which have the fame ratio to the fame, 

 or equal things, are equal ; and vice "verfd. 



6. If you multiply any quantities, as A and B, by the 

 fame, or equal quantities ; their produfts D and E will be 

 to each other as A and B. 



7. If you divide any quantities, as A and B, by the fame 

 or equal quantities, the quotients F and G will be to each 

 other as A and B. 



8. The exponent of a compound ratio is equal to the 

 faftum of the exponents of the fimple ratios. 



9. If you divide either the antecedents or the confequents 

 of fimilar ratios, A : B, and C : D, by the fame E ; in 

 the former cafe, the quotients F and G will have the fame 

 ratio to the confequents B and D ; in the latter, the ante- 

 cedents A and B will have the fame ratio to the quotients 

 H and K. 



10. If there be feveral quantities in the fame continued 

 ratio A, B, C, D, E, &c. the hrft A is to the third C, in 

 a duplicate ratio ; to the fourth D, in a triplicate ; to the 

 fifth E, in a quadruplicate, &c. ratio of the ratio of the firft 

 A, to the fecond B. 



11. If there be any feries of quantities in the fame ratio, 

 A, B, C, D, E, F, &c. the ratio of the firft A to the lalt F 

 is compounded of the intermediate ratios A : B, B : C, 

 C : D, D : E, E : F, &c. 



12. Ratios compounded of ratios, of which each is equal 

 to each other, are equal among themfelves. Thus the ratios 

 90 : 3 :: 960 : 32, compounded of 6 : 3 :: 4 : 2, and 

 3 : I.;: 12 : 4; and 5 : i :: 20 : 4. 



For other properties of fimilar or equal ratios, fee Pko- 

 iORTION. 5 



Ratios, ReduBiun hf. — It is obvious that there is a 

 variety of cafes in which the real ratio of two quantities may 

 be exprefled in terms too great to be applied to any ufeful 

 purpofe ; of which we have an example in the conilructioii 

 of planetariums, and fimilar ailronomical inllrumcnts. The 

 ratios of the times in which the feveral planets perform their 

 fidereal revolutions, are exprelled in very large numbers, far 

 exceeding the number of teeth that can be introduced into the 

 machinery ef a planetarium ; and it, therefore, becomes 

 necefliiry to find fmaller numbers, which, though they do 

 not cxprefs the true ratio, may approximate as near to the 

 truth as the ftate of the cafe will admit. Another inftance, 

 in which a reduction of the ratio of large numbers to 

 others exprefled in lower terms becomes neceflary, occurs 

 in the calendar ; for, according to the common reckoning, 

 the year is fuppofed to be 365 days, whereas it is known 

 to be nearly 365 days 6 hours ; it, therefore, becomes necef- 

 lary to have fome means of expreffing the ratio between the 

 true and the aifumed length of the years, in order that, by 

 a proper intercalation, we may preferve an uniformity in the 

 feafons, with reference to the months, as we fliould other- 

 wife find the (horteil day transferred to the middle of June, 

 and the longeft to the month of December. 



This reduction of ratios is bed performed by means of 

 continued Fractions, of which a flcetch is given under that 

 article, as alfo under the article Indeteuminate Analyfis, 

 but which we (hall probably treat at greater length in a 

 fupplement to the prefeiit work, on which account it is not 

 our intention to enter much into the rationale of the theory 

 in this place, but merely to (late the rules by which the 

 required reduftion is to be performed. 



To reduce a ratio exprejjed in large numbers, to others nearly 

 equivalent-, but reprejcnted injimpler terms. 



Rule I. — Divide the greater of the two numbers by the 

 lefs ; then the divifor by the remainder, and fo on, as in 

 finding the greatell common meafure of two numbers, and 

 referve the feveral quotients, which may be denoted by 

 a, b, c, d, &c. 



2. Write down the feveral quotients, thus ; a, b, c, d, e, &c. 

 from which the feries of converging fradlions or ratios will be 

 derived as follows ; vix. the firll fraction will have unity for 

 its numerator, and the firil quotient, a, for its denominator; 

 the fecond will have the fecond term, b, for its numerator, 

 and for its denominator ^i -|- i; and the numerators of all 

 the fucceeding fraftions will be found, by multiplying the 

 numerator lalt obtained, by the fucceeding quotient in the 

 above feries, and adding to the produA the preceding nume- 

 rator. And the denominators are obtained by precifely 

 the fame rule, merely changing the word numerator into 

 denominator. 



In this rule we have fuppofed the ratio to be lefs than i, 

 or the numerator lefs than the denominator ; if the deno- 

 minator be lefs than the numerator, it mull be reverfed, 

 making the numerator what we have called the denomi- 

 nator, and the denominator the numerator. 



The laft fraftion of this feries will be the fame as 

 the original fraftion propofed, and the others will be fo 

 many approximate or converging fraftions, each of which 

 will approach nearer to the original fraiflion than the pre- 

 ceding one, and nearer than any other fraftion whofe terms 

 are not expreffed by greater numbers. This rule may be 

 exhibited analytically as follows : 

 a 

 Let be the propofed fraftion, and a, b, c, &c. the 



quotients obtained by the divifions as above, then the con- 

 verging 



